Numbers and reality

Question

I have a question I don't really know how to formulate, so apologies for the cloudy mess. The topic is on the meaning of numbers, that is when I say 3, I am referring to, say, $3$ avocados, or $3$ inches. Still, if the third avocado has a large bite missing, am I supposed to say $3$ in the real meaning of an integer? Or $2.89$ inches, since the indicator is in the scope of the third inch. At $2.2$ it is already in the scope of three. May be integer numbers point out to a complete quantity of something. What about time, which is a continuum? So, mathematics seems a semplification of real cases, each with its nuance, but I feel there is a gap from reality as it is. Something like words with things. Could you clarify? Thanks!

Answer 1

As with all things in science, math can only approximate the real world. For each problem you are facing you can use a bunch of mathematical models and some of them work better for your specific problem than others.

As you pointed out, natural numbers are useful for counting things and to formulate problems, like:

How can you divide 20 children into 3 groups (sets), such that the differences in group sizes is minimal?

The answer wouldn't be $\frac{20}{3}$ children per group (that would be hideous). So natural numbers totally suffice here.

However, the problem might be different, if you want to divide a cake into three pieces of equal size (of equal volume). Usually you are able the slice the cake in any way you want. So it's among other things possible to get three pieces of the same size (of course in reality you cannot cut with such precision). This is different than before. Here we are instead using all the rational numbers to model our problem.

If you instead have a physics problem, like:

A particle is moving in a straight line. Given velocity $v(t)$ for every point $t$ in time and initial velocity $v_0 = v(0)$, determine the position of said particle at any time $t$.

then you will find, that rational numbers don't really suffice. Here we want to use real numbers as the solution to the problem can be found by solving a differential equation (with integration). These are methods, that do not work in the rational numbers.

There are also many other kinds of problems, that need other kinds of mathematics to describe them "properly". It always depends on the specific problem you are facing.

Answer 2

It seems as though you are asking "why mathematics exists". I think that any answer will just be an opinion since there is no true answer to such a question.

That being said, I feel that mathematics exists in order to attempt to provide structure so that the human mind can attempt to grasp the complexity of the universe. Everything in mathematics is based on rules that have been found (or created) over time by mathematicians.

So when you ask whether there are two or three avocados, you have to choose what you consider a whole avocado.

Answer 3

It seems that you are looking for the concept of quantity. A quantity is a numeral value and a unit (whereas a pure number has no unit). Quantities describe real things, whereas pure numbers are not "real." For example, "three avocados" or "three inches" are "real," but $3$ is not (for example, it doesn't occupy space or have mass.)

A discrete quantity is the magnitude of a countable set (one whose elements are “mutually separated and individually distinct”). Its numeral value is a natural number (“division into a quantity less than a unit cannot be considered”) and its unit is clear at the start. An example of a discrete quantity is “three boys.”

A continuous quantity is the magnitude of a “continuum” (“a continuous entity which can be divided into any number of smaller parts” such that “any two such entities can be combined into a larger one”). Its numeral value (a decimal or a fraction) and its unit “have not been determined a priori.” An example of a continuous quantity is “three dollars.”

(See my summary of Kô Ginbayashi's Theory of Quantity here.)